It is well known (see e.g. [1]}) that under very mild conditions it holds \( h_{M}-h^{*}=O_{\operatorname{P}}(M^{-1/2}). \) 
In their pioneering work [2]} (see also [3]}) showed that in the case of discrete random variable \( \xi , \)  the convergence of \( h_{N} \)  to \( h^{*} \)  can be much faster than \( M^{-1/2}, \)  making Monte Carlo method particularly efficient in this situation. Turn now to the discrete time optimal stopping problem:
\(V=\sup _{1\le \tau \le K} \operatorname{E}[Z_{\tau }],\) 
